3 Transience and recurrence of sets relative to certain time inhomogeneous diffusion processes

Some inequalities related to transience and recurrence of Markov processes and their applications

Daehong Kim^∗ and Yoichi Oshima^†

Abstract

For an irreducible symmetric Markov process on a, not necessarily compact, state space associated with a symmetric Dirichlet form, we give Poincar´e type inequalities. As an application of the inequalities, we consider a time inhomogeneous diffusion process obtained by a time dependent drift transformation from a diffusion process and give general conditions for the transience or recurrence of some sets. As a particular case, the diffusion process appearing in the theory of simmulated annealing is considered.

Keywords: Dirichlet forms, Poincar´e type inequality, Recurrent process, Simulated anneal- ing, Time inhomogeneous diffusion processes

Mathematics Subject Classification 2000: 60J60, 60J45, 60H30, 31C25

1 Introduction

Let X be a locally compact separable metric space andm a positive Radon measure on X with full support. Consider an irreducible regular Dirichlet form (E,F) onL²(X;m) and its associated m-symmetric Markov process M= (X_t, P_x) on X. M is called transient if there exists a strictly positive function g∈L¹(X;m) such that Rg(x) =E_x(^R₀^∞g(X_t)dt)<∞ for a.e.x ∈ X. M is called recurrent if it is not transient or, equivalently, if P_x(σ_F < ∞) = 1 q.e.x∈X for any non-exceptional setF inX, whereσ_F is the hitting time ofF.

Using the Dirichlet form, transience of M is characterized as follows : M is transient if and only if there exists a strictly positive functiong ∈L¹(X;m) and a constant k₁(g) such

that _Z

X|u(x)|g(x)dm(x)≤k1(g)E(u, u)^1/2, u∈ F (1) ([4]). As an L²-version of (1), the following result also holds (see [3],[12]): For any non- negative boundedm-integrable function g such thatkRgk∞<∞,

Z

X

u²(x)g(x)dm(x)≤2kRgk_∞E(u, u), u∈ F. (2)

∗This research was partially supported by Grant-in-Aid for Scientific Research No.17740062

†This research was partially supported by Grant-in-Aid for Scientific Research No.18540131

In particular, if kR1k_∞<∞, then (2) holds forg= 1 without the factor 2 in the righthand side. On the other hand, if M is Harris recurrent, it is known that there exists a strictly positive functiong∈L¹(X;m), a non-null setC ofX and a constantk₂(g) such that

Z

X|u(x)− hν_C, ui|g(x)dm(x)≤k₂(g)E(u, u)^1/2, u∈ F, (3) whereν_C(·) =m(·)/m(C) and hν_C, ui=^R_Xu(x)dν_C(x).

For a given set F ⊂ X, we say that F is a recurrent set of M if P_x(σ_F <∞) = 1 for a.e.x∈X. In this case, lim_T→∞P_x(σ_F ◦θ_T <∞) = 1 for a.e.x∈X. If this limit vanishes, then we callF a transient set of M.

In this paper, we consider some inequalities of Poincar´e type related to recurrent Markov processes and apply them to certain time inhomogeneous diffusion process to give general criteria for the transience and recurrence of some sets.

As a particular case, if we assume thatm(X)<∞and the generator ofMhas a spectral gap λ1 >0, then for anyλsuch that 0< λ≤λ1,

ku− hm, uik²2 ≤ 1

λE(u, u), u∈ F, (4)

wherek · kp denotes theL^p(X;m)-norm. In this case, the 1-resolventR1 of Msatisfies kR₁f − hm, fik2≤ 1

1 +λkf− hm, fik2, f ∈L²(X;m). (5) Note that the constant 1/(1 +λ) of the righthand side of (5) is less than 1.

In§2, instead of the existence of a positive lower bound of the spectral gap, we start from the assumption that

sup

x∈XkR₁(x,·)−m(·)k ≤2γ (6) for some γ < 1, where kνk denotes the total variation of the signed measure ν defined by kνk=ν(B⁺)−ν(B⁻) in terms of the Hahn decomposition X =B⁺∪B⁻ relative to ν. In this case, it is easy to see that

kR₁f − hm, fik2 ≤2γkf− hm, fik2, f ∈L²(X;m).

But the constant 2γ in the righthand side can be greater than one. Hence, it is not the optimal constant in the case ofL²(m)-setting. In Lemma 2.1, we show that the constant 2γ can be replaces by γ in the above inequality, that is,

kR1f − hm, fik2≤γkf− hm, fik2, f ∈L²(X;m). (7) This also shows that (1−γ)/γ is a lower bound of the spectral gap, that is,

Z

X

(u(x)− hm, ui)²dm(x)≤ γ

1−γE(u, u), u∈ F. (8)

Using this lemma, we shall also show anL²-version of (3) for general Harris recurrent Markov processes. Although the constant in (7) is sharper than that of (6), to discuss the estimates for any starting points, we need to use (6).

There are many interesting features concerning the transience or recurrence of some sets in the time inhomogeneous case because a set can be transient or recurrent depending on the fluctuation of the generator relative to the time parameter, unlike the time homogeneous case.

In§3, we consider the time inhomogeneous diffusion process M^ρ= (X_t, P_(s,x)^ρ ) associated with the family of energy forms (E^(t),F ∩L²(X;µ_t)) on L²(X;µ_t) defined by

E^(t)(ϕ, ψ) = 1 2

Z

X

ρ²(t, x)dµ_hϕ,ψi(x) (9)

with a strictly positive time dependent weight functionρ(t,·)∈ F, wheredµt(x) =ρ²(t, x)dm(x).

As a main result, we give some general criteria onρfor the transience or recurrence of some sets relative to M^ρ by applying the inequalities (2) and (8). As an example, we apply our criteria to a Brownian motion B on a compact connnected Riemannian manifold X and a weight function ρ(t, x) given by

ρ(t, x) = exp

³−(U(x)/c) log√ 1 +t

´

, c >0. (10)

Indeed, more profound properties of the diffusionB^ρcan be found in the theory of simulated annealing ([5],[6]).

2 Some inequalities related to transience and recurrence

As is stated in§1, some characterizations of transience and recurrence of symmetric Dirichlet forms (E,F) on L²(X;m) are given in Fukushima et.al. ([4]). The transience of (E,F) is characterized by (1). Furthermore, in this case, anL²-version (2) holds.

The purpose of this section is, after getting the inequality (8) for the Markov processes satisfying the inequality (6), to show anL²-version of (3) for general Harris recurrent Markov processes. To show the inequality (8), we make the following assumptions onM.

(A) Mis recurrent and there exists γ <1 such that (6) holds.

In this case, for any n≥1,

kR₁ⁿ(x,·)−m(·)k ≤2γⁿ, x∈X (11) ([11]). Note that (11) implies thatmis a probability measure. The condition(A)is satisfied if X is compact and R₁ is strong Feller, or more generally, if the density of the absolutely continuous part ofR1(x,·) relative tomis bounded from below by a positive constant ([11]).

LetX =B(x)⁺∪B(x)⁻be a Hahn decomposition relative to the signed measureR(x,·)− m(·). Then

kfsupk∞≤1

k(R1−m)fk_∞ = sup

x∈X

³

(R1−m)I_B(x)⁺ −(R1−m)I_B(x)−

´

= sup

x∈XkR₁(x,·)−m(·)k ≤2γ.

Hence, using the same symbolk·kpto represent the operator norm ofR₁−minL^p(X;m), we havekR₁−mk_∞≤2γ. For anyf ∈L¹(X;m), put ¯f =f−hm, fiandB_f ={x:R₁f¯(x)≥0}. Then, by the symmetry ofR₁, we see

k(R1−m) ¯fk1

= Z

Bf

R₁f¯(x)dm(x)−^Z

X\Bf

R₁f¯(x)dm(x)

= Z

X

f¯(y){(R₁(y, B_f)−m(B_f))−(R₁(y, X\B_f)−m(X\B_f))}dm(y)

≤ sup

y∈XkR1(y,·)−m(·)k · kf¯k1

≤ 2γkf¯k1

and thuskR1−mk1≤2γ. Denote the total variation measure |R1(x,·)−m(·)|by

|R1(x,·)−m(·)|(A)

=^¡R₁(x, A∩B(x)⁺)−m(A∩B(x)⁺)^¢−^¡R₁(x, A∩B(x)⁻)−m(A∩B(x)⁻)^¢. Then the operator norm onL¹(X;m) determined by|R1(x,·)−m(·)|coincides withkR1−mk1. By a similar argument using (11) instead of (6), we have

kRⁿ₁ −mk∞≤2γⁿ. (12) Let denote (·,·)_µ the inner product onL²(X;µ).

Lemma 2.1 Suppose thatM satisfies the assumption (A). Then (7) and (8) hold.

Proof. Put

λ₁ = inf

½ E(u, u) ku− hm, ui|²m

:u∈ F

¾ .

By using the spectral representation−G =^R₀^∞dE_λ of the generator G of Mand (12), µ 1

1 +λ₁

¶_n

≤ inf

½Z _∞

λ1

µ 1 1 +λ

¶_n

d(E_λu, u) :kukm= 1,hm, ui= 0

¾

= k(R1)ⁿ−mk²_m≤ k(R1)ⁿ−mk²₀ ≤2γⁿ.

Since the righthand side tends to zero asn→ ∞, it follows thatλ1 >0 and 1+λ1 ≥2^−1/nγ⁻¹ and henceλ1 ≥(1−γ)/γ. (7) and (8) follow easily from this. 2

Define a potential kernel K by Kf(x) =

X∞ n=1

(Rⁿ₁f(x)− hm, fi). (13) By virtue of (11),Kf(x) is well defined for all x and satisfies

kKfk∞≤2 X∞ n=1

γⁿkfk∞= 2γ

1−γkfk∞ (14)

for allf ∈L^∞(X;m). Similarly, by using (??), for allf ∈L²(X;m), (f, Kf)m =

X∞ n=1

( ¯f ,(Rⁿ₁ −m) ¯f)m≤ ^X∞

n=1

γⁿkf¯k²2 = γ

1−γkf¯k²2. (15) foru∈L²(X;m). By using the representation

((Rⁿ₁ −m)f, g)_m = Z _∞

0

e^−t(Φnf(t,·),Φng(t,·))_mdt

for Φ_2kf(t, x) = (R^k₁ −m)f(x) and Φ_2k+1f(t, x) = (R^k₁−m)p_t/2f(x), we have from (15) (Kf, Kf)_m =

X∞ n=1

((Rⁿ₁ −m)f, Kf)_m

= X∞ n=1

Z _∞

0

e^−t(Φ_nf(t,·),Φ_n(Kf)(t,·))_mdt

= X∞ n=1

Z _∞

0

e^−t(Φnf(t,·), KΦnf(t,·))_mdt

≤ γ

1−γ X∞ n=1

Z _∞

0

e^−t(Φnf(t,·),Φnf(t,·))_mdt

= γ

1−γ (f, Kf)_m. (16)

Similarly to (16), the potential kernelK⁽⁰⁾f =Kf +f − hm, fi=^P∞_n=0(Rⁿ₁ −m)f satisfies

³

K⁽⁰⁾f, K⁽⁰⁾f^´

m≤ 1 1−γ

³

f, K⁽⁰⁾f^´

m. (17)

Next, we shall consider the case that the condition (A) is not necessarily satisfied. We assume thatMis recurrent in the sense of Harris, that is, for any F ⊂X with m(F)>0,

Z _∞

0

I_F(Xt)dt=∞ a.s. Px for all x∈X.

In particular, if M is recurrent and R1(x,·) is absolutely continuous relative to m for all x∈ X, then M is recurrent in the sense of Harris ([2],[8]). In this case, as we stated in §1,

the inequality (3) holds. Now, let us assume thatM satisfies Harris recurrence condition to derive anL²-version of (3).

Define for any positive continuous additive functionalA_t=^R₀^tI_C(X_s)ds, the kernels R_A^α andK_A^α by

R^α_Af(x) = Ex

µZ _∞

0

e^−αt−Atf(Xt)dt

¶ , K_A^αf(x) = E_x

µZ _∞

0

e^−αAtf(X_t)dA_t

¶ .

In particular, put R_A = R_A⁰. Under the present assumption of Harris recurence, for a set C with m(C) > 0, K_A^α is the resolvent of the recurrent time changed process on C by A_t. Furthermore, we can chooseC satisfying 0< m(C)<∞ and

sup

x∈C

°°°K_A¹(x,·)−νC(·)^°°°≤2γ

for someγ <1, whereνC = (1/m(C))m|C ([9],[11]). Similarly to (11), it then holds that

°°°(K_A¹)ⁿ(x,·)−νC(·)^°°°≤2γⁿ, ∀n≥1. (18)

By virtue of Lemma 2.1, for anyf ∈L²(C;ν_C),

k^³(K_A¹)ⁿ−ν_C^´fkm≤γⁿkf− hν_C, fik2. (19) Put

KA= X∞ n=1

³

(K_A¹)ⁿ−νC

´

, K_A⁽⁰⁾= X∞ n=0

³

(K_A¹)ⁿ−νC

´ . From the symmetry,

hν_C, R_Afi=

³ K_A¹1, f

´

m =hm, fi. Define a kernelK by

Kf =K_A⁽⁰⁾R_Af =K_A¹K_A⁽⁰⁾R_Af+R_Af − hm, fi. By using the Markov property,

αR_αK_A¹h = R_α(I_C ·K_A¹h) +K_A¹h−R_α(I_Ch), αRαRAh = Rα(IC·RAh) +RAh−Rαh.

SinceK_A¹K_A⁽⁰⁾g=K_A⁽⁰⁾g−g+hν_C, gi, we have αR_αK_A¹K_A⁽⁰⁾R_Af

= Rα(IC·K_A¹K_A⁽⁰⁾RAf) +K_A¹K_A⁽⁰⁾RAf −Rα(IC·K_A⁽⁰⁾RAf)

= Rα(I_C·K_A⁽⁰⁾R_Af−I_C·R_Af+I_C· hν_C, R_Afi) + K_A¹K_A⁽⁰⁾R_Af−R_α(I_C ·K_A⁽⁰⁾R_Af)

= −Rα(IC ·RAf) +RαIC · hm, fi+K_A¹K_A⁽⁰⁾RAf.

Hence it holds that

(I−αR_α)Kf =R_αf−R_αI_C· hm, fi (20) and consequently,

K(I−αRα)f =Rαf − hνC, Rαfi. (21) Moreover, since

Z

X

³ K_A¹h

´₂

(x)g(x)dm(x) ≤ Z

X

K_A¹h²(x)g(x)dm(x)

= Z

X

RAg(x)h²(x)dνC(x)

≤ kRAgk_∞^Z

X

h²(x)dνC(x),

applying (17) to K_A⁽⁰⁾|C×C, we get for any bounded non-negative function g ∈ L¹(X;m) satisfying kR_Agk_∞<∞ that

(K_AR_Af, K_AR_Af)_g·m = ^°°_°K_A¹K_A⁽⁰⁾R_Af^°°_°²

L²(g·m)

≤ kR_Agk∞

³

K_A⁽⁰⁾R_Af, K_A⁽⁰⁾R_Af

´

νC

≤ kRAgk∞ 1 1−γ

³

RAf, K_A⁽⁰⁾RAf

´

νC

= kRAgk_∞ 1 1−γ

³

f, K_A¹K_A⁽⁰⁾RAf

´

m

= kRAgk_∞ 1

1−γ (f, KARAf)_m.

Note that R_A is the potential kernel of the transient Dirichlet form EA = E + (·,·)ν_C on L²(X;m) andR_AI_C = 1. Thus we have from (2),

Z

X

(R_Af− hm, fi)²(x)g(x)dm(x)

= Z

X

(R_A(f− hm, fi ·I_C))²(x)g(x)dm(x)

≤2kRAgk∞E(RA(f − hm, fi ·IC), RA(f− hm, fi ·IC))

≤2kR_Agk_∞EA(R_A(f − hm, fi ·I_C), R_A(f − hm, fi ·I_C))

= 2kRAgk_∞(f − hm, fi ·IC, RA(f − hm, fi ·IC))_m

≤2kR_Agk_∞(f, R_A(f − hm, fi ·I_C))_m. Therefore

(Kf, Kf)_g·m

= (KARAf+RAf− hm, fi, KARAf +RAf− hm, fi)_g·m

≤2 (KARAf, KARAf)_g·m+ 2 Z

X

(RAf(x)− hm, fi)²g(x)dm(x)

≤4kRAgk∞

½ 1

1−γ (f, KARAf)_m+ (f, RAf− hm, fi)_m

¾

≤ 4

1−γkRAgk∞(f, Kf)m

= 4

1−γkRAgk∞E(Kf, Kf). (22)

On the other hand, from (20) and (21),

Kf =R1Kf+R1(f− hm, fi ·IC) and

R₁f =Kf−KR₁f +hν_C, R₁fi.

Thus the images ofK and R1 coincide except a difference of a constant factor which makes the integral byνC zero. Hence, we have from (22),

Z

X|R1f(x)− hνC, R1fi|²g(x)dm(x)

= Z

X|K(I−R₁)f(x)|²g(x)dm(x)

≤ 4

1−γkR_Agk_∞E(K(I −R₁)f, K(I −R₁)f)

= 4

1−γkR_Agk_∞E(R₁f, R₁f).

Further, approximatingu∈ F by a sequence of functions of the formR₁f as in the proof of Lemma 2.1, we get the following result.

Theorem 2.1 If M is recurrent in the sense of Harris, then there exists a set C such that 1

m(C) Z

X|u(x)− hν_C, ui|²g(x)dm(x)≤ 4kRAgk∞

1−γ E(u, u) (23)

for anyu∈ F and a bounded non-negative functiong∈L¹(X;m) such that kR_Agk_∞<∞.

3 Transience and recurrence of sets relative to certain time inhomogeneous diffusion processes

In this section, we assume that we are given an irreducible m-symmetric diffusion process M= (Xt, Px) on X which is associated with the Dirichlet form (E,F) given by

E(ϕ, ψ) = 1 2

Z

X

dµ_hϕ,ψi(x). (24)

We consider that the path space is canonical and X_t(ω) = ω(t). Denote the associated generator by G. Fix a strictly positive continuous function ρ(t, x) such that ρ(t,·) ∈ F and t 7→ ∂ρ(t,·)/∂t is a measurable function from [0,∞) to L²_loc(X;m). Put µ_t(dx) = ρ²(t, x)m(dx) and consider the Dirichlet form (E^(t),F^(t)) on L²(X;µ_t) determined by

E^(t)(ϕ, ψ) = 1 2

Z

X

ρ²(t, x)dµ_hϕ,ψi(x). (25) Denote by G^(t) the generator corresponding to (E^(t),F^(t)). A time inhomogeneous diffusion processM^ρ = (Xt, P_(s,x)^ρ ) is said associated with the family of Dirichlet forms (E^(t),F^(t)) if its transition functionut(s, x) =E_(s,x)^ρ (ϕ(Xt−s)) satisfies the terminal value problem

∂ut(s, x)

∂s +G^(s)ut(s, x) = 0, ut(t, x) =ϕ(x), (26) fors < t. Denote by R^ρ_α the resolvent ofM^ρ, that is

R^ρ_αϕ(s, x) =E_(s,x)^ρ µZ _∞

0

e^−αtϕ(X_t)dt

¶ . Then (26) is equivalent to

−

µ∂R^ρ_αϕ(s,·)

∂s , ψ

¶

µs

+Eα^(s)(R^ρ_αϕ(s,·), ψ) = (ϕ, ψ)µs (27) for anyϕ∈ F^(s).

There also exists a diffusion process M^cρ = (X_s,P^b_(t,y)^ρ ) which is a dual process of M^ρ in the sense _Z

X

E_(s,x)^ρ (ψ(X_t−s))ϕ(x)dµ_s(x) = Z

X

Eb_(t,y)^ρ (ϕ(X_t−s))ψ(y)dµ_t(y) (28) for any ϕ, ψ ≥ 0. Note that the measure P^bρ is not necessarily sub-Markov. In fact, P^bρ is given by the following transformation by a multiplicative functional

Pb_(t,y)^ρ (Λ) =E_y µ

exp µ

M_t^[log_−s^ρ]−1

2hM^[logρ]it−s

¶

e^−Bbt−s : Λ

¶ ,

for Λ∈σ(Xτ;τ ≤t−s), whereMτ^[logρ]is the martingale part appearing in the decomposition logρ(t−τ, X_τ)−logρ(t, X₀) =M_τ^[logρ]+N_τ^[logρ]

into a martingale additive functional of finite energy and a continuous additive functional of zero energy relative toP^b_(t,y)^ρ , and

Bbs= Z _s

0

∂logρ

∂t (t−τ, Xτ)dτ.

Hence, we have

Pˆ_(t,y)^ρ (X)≤exp (`<sub>t</sub>(s)), `_t(s) = Z _t−s

0

°°°°∂logρ

∂t (t−τ,·)^°°_°°

∞dτ. (29) Fix a closed set F of X such that ρ²(t,·) ∈ L¹(D;m) for D = X\F. By considering ρ²(t, x)/Z(t) instead of ρ²(t, x), we may assume that µ_t(dx) =ρ²(t, x)m(dx) is a probability measure onD, where

Z(t) = Z

D

ρ²(t, x)dm(x).

Let f be a non-negative function on D such that hµ_s, fi = 1 for fixed s ≥ 0. For such f, define the functionub^D_s by

b

u^D_s(t, y) =E^b_(t,y)^ρ (f(X_t−s) :t−s < σ_F), y∈D (30) and put

Hb_s^D(t) = Z

D

(ub^D_s )²(t, y)dµt(y).

We assume that the number

λ_D(t) =− inf

x∈D

∂

∂tlogρ²(t, x)

is finite. For instance, this assumption holds if D is relatively compact. Then we have the following lemma relative toH^b_s^D(t).

Lemma 3.1 (i) For anys < t,

E^(t)(ub^D_s(t,·),ub^D_s(t,·))≤ −1 2

d

dtH^b_s^D(t) +1

2λ_D(t)H^b_s^D(t).

(ii) If limt→∞Hb_s^D(t) = 0, then P_f^ρ_·µs(σF <∞) = 1.

Proof. (i) Sinceub^D_s satisfies 1 ρ²(t, y)

∂(ρ²ub^D_s)

∂t (t, y) =G^b(t)ub^D_s(t, y) (31) with condition

b

u^D_s(s, y) =f(y), ub^D_s(t, y) = 0 for y∈F, by multiplyingub^D_s (t, y) and integrating onD bydµ_t(y), we have

Z

D

∂(ρ²ub^D_s(t, y))

∂t u^bD_s (t, y)dm(y) =−E^(t)³ub^D_s (t,·),ub^D_s(t,·)

´

. (32)

Since the lefthand side of (32) can be written as 1

2 d dt

Z

D

(ub^D_s )²(t, y)ρ²(t, y)dm(y) +1 2

Z

D

(ub^D_s)²(t, y)∂ρ²(t, y)

∂t m(dy)

we get the result.

(ii) By virtue of the duality relation (28), it holds that P_f^ρ_·µ

s(t−s < σ_F) = E_µ^ρ_s(f(X₀)I_D(X_t−s) :t−s < σ_F)

= E^b_µ^ρ_t(f(Xt−s)ID(X0) :t−s < σF)

= Z

D

b

u^D_s (t, y)dµt(y)

≤ ^qH^b_s^D(t).

Hence, we have the assertion. 2

Assume that F is a non-exceptional closed set. By virtue of the irreducibility of M, its part processMD on D is transient. Hence, applying (2) for MD, for any bounded positive functiong∈L¹(X;m) such thatkR^Dgk_∞<∞,

Z

D

u²(x)g(x)dm(x)≤2kR^Dgk∞ E(u, u) (33) for all u ∈ FD = {u ∈ F : ˜u = 0 q.e.on F}. If m(D) < ∞ and kR^D1k_∞ < ∞, then (33) holds forg = 1. As a typical case, this holds ifD is compact and the transition functionp_t ofM is strong Feller. In fact, it then holds that inf_x∈Dp_t(x, F)>0 and

p^D_t 1(x)≤1−p_t(x, F)≤1− inf

x∈Dp_t(x, F)<1 for anyx∈D, wherep^D_t is the transition function of M_D.

Now, we give a general criterion on ρfor the recurrence of the set F relative toM^ρ. Put δD(t) = inf_Dρ²(t,·)

sup_D(ρ²(t,·)/g(·)). Sinceg is bounded, δ_D(t)<∞.

Theorem 3.1 Suppose that there exists a positive function g∈L¹(D;m) such that

Tlim→∞

Z _T

s

µ

λ_D(t)−^°°°R^Dg^°°_°⁻¹

∞ δ_D(t)

¶

dt=−∞. (34)

ThenP_f^ρ_·µ

s(σ_F <∞) = 1 for any non-negative function f with hµ_s, fi= 1. In particular, if the transition density of M^ρ exists, then P_(s,x)^ρ (σ_F <∞) = 1 for all x∈D.

Proof. Letg >0 be a function satisfying the stated condition. Then we have from (33) Hb_s^D(t) ≤ 2 sup

x∈D

(ρ²(t, x)/g(x))kR^Dgk∞ E^³u^bD_s(t,·),ub^D_s (t,·)

´

≤ 2kR^Dgk_∞ δ_D⁻¹(t) E^(t)³ub^D_s (t,·),ub^D_s(t,·)^´.

Combining this with the result (i) of Lemma 3.1, we get that

°°°R^Dg^°°°⁻¹

∞ δD(t) H^b_s^D(t)≤ −d

dtH^b_s^D(t) +λD(t)H^b_s^D(t), that is,

d

dtlogH^b_s^D(t)≤ µ

λ_D(t)−^°°°R^Dg^°°_°⁻¹

∞ δ_D(t)

¶ . Hence, we have

Hb_s^D(T)≤H^b_s^D(s) exp ÃZ T

s

µ

λ_D(t)−^°°°R^Dg^°°_°⁻¹

∞ δ_D(t)

¶ dt

!

. (35)

Therefore the first assertion follows from Lemma 3.1 (ii). If the transition densityp^ρ(s, x;t, y) ofM^ρ exists, then

P_(s,x)^ρ (t−s < σ_F) = P_(s,x)^ρ (τ −s < σ_F, t−τ < σ_F ◦θ_τ−s)

= P_f^ρ_·µτ (t−τ < σF)→0, t→ ∞

forf(y) =p^ρ(s, x;τ, y). 2

Example 3.1 Suppose that X is a Riemannian manifold with volume element m and B = (Xt, Px) the Brownian motion on X. Then the associated Dirichlet form (E,F) on L²(X;m) is given by

E(ϕ, ψ) = 1 2

Z

X∇ϕ(x)· ∇ψ(x)dm(x).

LetU be a smooth locally bounded non-negative function onX andρ(t, x) a function defined by

ρ(t, x) = exp µ

−1

2β(t)U(x)

¶

. (36)

forβ(t) = (1/c) log(1 +t), c >0. We consider the associated time inhomogeneous diffusion processB^ρ= (Xt, P_(s,x)^ρ ) for the function (36).

Fix a connected componentDof a level set of the form{x:U(x)≤b}and leta= infDU. To makeµt(D) = 1, we considerρ²(t, x)/Z(t) instead of ρ²(t, x), that is we consider as

µ_t(dy) = ρ²(t, y)m(dy)

Z(t) , Z(t) = Z

D

ρ²(t, y)dm(y).

Since D is compact, it is easy to see that the part process B_D of B on D satisfies (33) for g= 1. By elementary calculations, since

λ_D(t) = b

c(1 +t) + d

dtlogZ(t), and δ_D(t) = (1 +t)^−(b−a)c , (37)

we have

Z _T

s

λD(t)dt = b

clog 1 +T

1 +s + log Z(T) Z _T Z(s)

s

δ_D(t)dt = c c−(b−a)

³

(1 +T)^{1−(b−a)/c}−(1 +s)^{1−(b−a)/c}

´

and for anyε >0,

m({a < U < a+ε})(1 +t)^−a+εc ≤Z(t)≤m(D)(1 +t)^−ac. Therefore, ifb−a < c,

Tlim→∞

Z _T

s

µ

λ_D(t)−2^°°_°R^D1^°°_°⁻¹

∞ δ_D(t)

¶

dt=−∞

and which implies that the set F = X \D is a recurrent set relative to B^ρ by virtue of Theorem 3.1.

Next, we turn to a general condition on ρ for the transience of some sets relative toM^ρ. We assume that the state space X is compact and M is recurrent on it. Before considering time inhomogeneous process M^ρ, we give an estimation of the type (??) relative toM^ρ¯ for time independent ¯ρ.

Fix a positive function ¯ρ(x) ∈ D(G) such that µ_{log ¯ρ}(x) := dµ_{hlog ¯ρ}2,log ¯ρ²i(x)/dm(x) <∞ and kGlog ¯ρk_∞ < ∞. Put d¯µ(x) = ¯ρ²(x)dm(x). Let (E^ρ¯,F^ρ¯) be the Dirichlet form on L²(X; ¯µ) determined by

E^ρ¯(ϕ, ψ) = 1 2

Z

X

¯

ρ²(x)dµ_hϕ,ψi(x)

andM^ρ¯= (X_t^ρ¯, P_x^ρ¯) the associated diffusion process with resolventR^ρ_α^¯. Puth(x) =Glog ¯ρ(x)+

2µ_{log ¯ρ}(x), C_t=^R₀^th⁺(X_s)ds and

R^α_Cf(x) =Ex

µZ _∞

0

e^−αt−Ctf(Xt)dt

¶ . Then it satisfies

R^α_Cf(x) =Rαf(x)−R_C^α(h·Rαf)(x) (38) (§4.6 in [4]). If the density R_α(x, y) of R_α(x, dy) relative to m(dy) exists, then (38) implies that the density ofR^α_C(x,·) relative tom also exists. We denote its density byR^α_C(x, y).

For convenience, we will assume sup_x∈Xρ(x) = 1 with no loss in generality. Put¯ H^ρ¯(x, y) = inf

η sup

0≤t≤1

³−log ¯ρ²(η(t))

´

, x, y∈X, whereη(t),0≤t≤1 is a curve in X connecting x and y. Furthermore, put

m( ¯ρ) = sup

x,y∈X

n

H^ρ¯(x, y) + log ¯ρ²(x) + log ¯ρ²(y) o

, γ( ¯ρ) = 1−e^{−m( ¯ρ)/2}inf

z,wR¹_C(z, w).